Mathematical & Logical Puzzles -Q1

1.   The Flight Around The World

            A group of aeroplanes is based on a small island.  The tank of each plane holds just enough fuel to take it half-way around the world.  Any desired amount of fuel may be transferred from the tank of one plane to the tank of another whilst the planes are in flight.  The only source of fuel is on the island,  and for the purposes of this problem it is assumed that there is no time lost in refuelling either in the air or on the ground.

            What is the smallest number of planes that will ensure the flight, non-stop, of one plane around the world on a great circle assuming that each of the planes has the same constant ground speed and rate of fuel consumption, and that all planes return safely to their island base?

2.   Who Is The Engineer?

            Smith, Jones and Robinson are engineer, brakeman and fireman on a train but not necessarily in that order.  Also riding the train are three passengers with the same three surnames, to be identified in the following premises with a “Mr.” before their names.

·      Mr. Robinson lives in Los Angeles.

·      The brakeman lives in Omaha.

·      Mr. Jones long ago forgot all the algebra he ever learnt at High School.

·      The passenger with the same name as the brakeman lives in Chicago.

·      The brakeman and one of the passengers, a distinguished mathematical physicist, attend the same church.

·      Smith beat the fireman at billiards.

What is the name of the engineer?

3.   The Coloured Hats

            

Three men, Alan, Brian and Colin are blindfolded and are told that either a red hat or a green hat will be placed on each of their heads.  After this is done, the blindfolds are removed and the men are asked to do the following:

·      To raise a hand if they see a red hat.

·      To leave the room as soon as they are sure of the colour of their own hat.

            All the hats happen to be red, so all the men raise a hand.  Several minutes elapse until Colin, who is more astute than the others, leaves the room.  How did he deduce the colour of his hat?

4.   The Coloured Hats Again

        

    At a party, four people played a game.  Three of them sat one behind the other, so that Arthur saw Bill and Colin, and Bill saw only Colin who sat in front and saw nobody.

            Dave had five hats which were showed to his three friends.  Three of the hats were blue and two were red.  Now Dave placed a hat on the head of each of his three friends, putting aside the remaining two hats.

            He now asked Arthur what colour his hat was.  Arthur said he couldn’t tell.  Bill, asked the colour of his own hat didn’t know for certain either.  Colin, however, although he couldn’t see any hat at all, gave the correct answer when asked what the colour of his hat was!  What was the colour of Colin’s hat and how did he deduce it?

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